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We consider Gaussian random waves on hyperbolic spaces and establish varianceasymptotics and central limit theorems for a large class of their integralfunctionals, both in the high-frequency and large domain limits. Our strategyof proof relies on a fine analysis of Wiener chaos expansions, which in turnrequires us to analytically assess the fluctuations of integrals involvingmixed moments of covariance kernels. Our results complement several recentfindings on non-linear transforms of planar and arithmetic random waves, aswell as of random spherical harmonics. In the particular case of 2-dimensionalhyperbolic spaces, our analysis reveals an intriguing discrepancy between thehigh-frequency and large domain fluctuations of the so-called fourthpolyspectra -- a phenomenon that has no counterpart in the Euclidean setting.We develop applications of a geometric flavor, most notably to excursionvolumes and occupation densities.

Nonlinear Functionals of Hyperbolic Random Waves: the Wiener Chaos Approach